Basis Vectors

We can choose 4 basis vectors in 3D space:

We then can find a transformation of any 4 points on 2D into basis vectors.

1. To do that, let add the third dimension to every 2D point:

(x,y) -> (x,y,1)

2. Now given 4 vectors in 3D space:

(x1,y1,1)
(x2,y2,1)
(x3,y3,1)
(x4,y4,1)

we want to find such matrix M that will transform basis vectors to our vector points.

3. We can call this matrix M base to points transform:

M * (0,0,0) = (x1,y1,1)
M * (0,1,0) = (x2,y2,1)
M * (0,0,1) = (x3,y3,1)
M * (1,1,1) = (x4,y4,1)

4. Then inverse matrix M⁻¹ will convert basis vectors back to points:

M⁻¹ * (x1,y1,1) = (0,0,0)
M⁻¹ * (x2,y2,1) = (0,1,0)
M⁻¹ * (x3,y3,1) = (0,0,1)
M⁻¹ * (x4,y4,1) = (1,1,1)